Everywhere between trivial and not-in-this-millennium:
Has memoization ever popped up on the wish-list?
It would, for example, be useful often for programs
for OEIS sequences.
(I guess this would a require a prohibitive amount of work)
Adjustable array starts: as soon as the operations on
indices are of additive nature (which may be the vast
majority for certain users), one may wish for the first
element being at index zero. If the start index is made
adjustable, one might just make it fully adjustable.
(Again, very limited hope this will ever make it).
Multivariate polynomial expansion and factorization:
factorization is the one functionality missing that
let's me fire up Sage occasionally.
Is it on the list?
(Yes, I did read
http://pari.math.u-bordeaux1.fr/faq.html#polmultivar )
Passing by reference:
Well, this would just be hugely useful.
I'd be able to create many (very efficient) combinatorial
generators for Pari (essentially all that's in the FXT-lib's
combinatorial department) without much effort.
(No idea if this is reasonably doable)
And just to draw some serious flames:
Something like (C++) classes.
Hypergeometric series (not functions, that would
be way too much work!), see http://jjj.de/pari/hyper2cf.gpi
(looks doable to me)
q-stuff: q-pochhammer, q-binomial, q-factorial, q-multinomial, ...
Again the "series" part should be _much_ simpler than
the "function" part (which I do not even dare to suggest).
See e.g. http://jjj.de/pari/qbin.gpi
[Similarly, but more specialized:
For eta (with extra arg), theta{2,3,4}, Ramanujan-functions,
Eisenstein series, ell-lambda, and more see
http://jjj.de/pari/eta.gpihttp://jjj.de/pari/theta.gpi
Lambert series, see
http://jjj.de/pari/lambert.gpi
Some q-trigonometry can be found at
http://jjj.de/pari/qtrig.gpi (reads the following)
http://jjj.de/pari/qtrig1.gpihttp://jjj.de/pari/qtrig1r.gpihttp://jjj.de/pari/qtrig2.gpihttp://jjj.de/pari/qtrig2r.gpi
]
SVD, see http://jjj.de/pari/matsvd.gpi
Other matrix decompositions: Schur, what else?
(SVD should be doable).
Circulant and Toeplitz matrices, see
http://jjj.de/pari/matcirc.gpi
(trivial, right?)
Pade approximant for series, see
http://jjj.de/pari/pade.gpi
(looks doable)
An option to have polynomials printed "constant first".
This is much neater in some cases, e.g., for rational
generating functions.
Option to have Vec(series) _not_ drop initial zeros.
An option (default on) that detects when somebody
is about to work on a god-awful contrived sequence
he/she wants to submit to the OEIS, that, upon such
detection, prints "You are wasting your time" and
switches the computer off. Un-setting this option
should be excruciatingly tedious.
Now this is all I could collect within 60 minutes :^)
Best regards, jj <--= potential flames go here